In yesterday's blog entry, I showed the importance of the first set of fundamental polarities, i.e. external and internal, which in a dynamic complementary manner conditions all phenomenal experience of life.

I then indicated how appreciation of such complementarity, that dramatically unfolds during Level 1 (i.e. The Circular Level) of Band 3, leads to a distinctly new type of scientific (and mathematical) appreciation that is of a qualitative - rather than quantitative - nature.

I customarily refer to these as Holistic Science and Holistic Mathematics respectively.

However during Level 2 (The Point Level) of Band 3 the additional unfolding of the second set of (fundamental) polarities also takes place in a dramatic fashion.

This second set relates to the dynamic relationship as between wholes and parts. This also manifests itself as the relationship (in any relevant context) as between general and particular, collective and individual, "higher" and "lower", analytic and holistic, qualitative and quantitative etc.

In conventional understanding, a great deal of reductionism is involved with respect to this relationship. Thus in scientific and - especially - mathematical terms, the qualitative aspect is grossly reduced in merely quantitative fashion. This leads to the accepted common sense notion of the whole (in any context) as representing the (quantitative) sum of its various parts.

Indeed the very meaning of "analytic" as applied to scientific interpretation, directly implies this reduced notion of the relationship of wholes and parts.

The key underlying factor for this reduced understanding is the attempt in explain reality in a merely conscious manner (that is directly identified with the linear use of reason).

However in the dynamics of understanding, the unconscious necessarily interacts with the conscious aspect. This implies the corresponding interaction of (holistic) intuition and (analytic) reason in the dynamic interchange as between wholes and parts (and in reverse manner, parts and wholes).

In the context of the "higher" contemplative development of Band 3, understanding is now sufficiently refined. that one can properly interpret, in scientific manner, this key relationship between wholes and parts.

I have mentioned before how at this time I developed a strong resonance with Jungian psychology, and it was indirectly, through consideration of his treatment of the 4 basic functions, that I could resolve the issue with an exciting new clarity.

Jung - though not specifically addressing the issue in this manner - used language and concepts that readily lend themselves to holistic mathematical interpretation.

So he described two rational functions, directly of a conscious nature, i.e. thinking and feeling and two irrational functions, directly of an unconscious nature, i.e. sense and intuition.

Now I had already come to interpret the first set of polarities (external and internal) as the holistic mathematical interpretation of the two roots of 1. In other words, external and internal represent poles that are positive (+ 1) and negative ( – 1) with respect to each other (which continually switch in a dynamic experiential fashion).

I now realised that the combined two sets of polarities could be interpreted as the holistic equivalent of the four roots of 1. So I begin to see that conscious and unconscious, in the dynamic switching between wholes and parts (and whole and parts), are in holistic mathematical terms "real" and "imaginary" with respect to each other.

There is always a degree to which polarities are relatively independent and also relatively interdependent with respect to each other.

For example when one identifies an object as external (in a conscious rational manner) external and internal are therefore necessarily separated (with respect to such recognition).

However equally to a degree, one is implicitly aware of the necessary interaction between poles, which assumes a common shared relationship. And recognition of the shared identity of both poles is of an unconscious intuitive nature!

In direct terms, this unconscious fusion remains hidden from consciousness (in a nondual fashion). However, indirectly, it projects itself into experience in an "imaginary" fashion (which precisely represents the holistic mathematical notion of "imaginary").

Thus, when we properly view experience, as entailing the interaction of both conscious and unconscious aspects, all experience is therefore of a complex nature (in holistic mathematical terms).

For example if one plans to buy a new house, one can identify this object in a conscious (i.e. real) manner. However equally - perhaps to a considerable - degree, the house will embody a deeper holistic aspect (ultimately reflecting the hidden unconscious desire for meaning). And this latter aspect represents the imaginary (indirectly conscious) aspect.

This is equally true of mathematical experience. A number theorist may indeed believe that the symbols used, can be interpreted in a real (i.e. rational) manner. However the very pursuit of this type of knowledge, will inevitably reflect a deeper holistic desire for meaning. So the imaginary aspect thereby becomes necessarily embodied in all the abstract symbols used, thus colouring the direction - and indeed interpretation - of mathematical truth in many ways (which the participant may not readily acknowledge).

So in the switching from wholes to parts (and in reverse manner part to whole), a decisive imaginary aspect is involved.

In psychological terms this entails the fundamental switch from concepts to perceptions and perceptions to concepts (which underlies all experience).

Let us take the - apparently - simple case of recognition of the number "2".

Now "2" (as in two objects) represents a number perception that is identified in a real conscious manner. Though it is referred to as an integer, it really represents in quantitative terms a part notion where "2" can be identified as part of a larger number collection.

However the perception of "2" has no meaning in the absence of the concept of number, which potentially applies to all numbers. This potential notion, that is infinite, is unconscious in nature and directly grasped through intuition. However indirectly it enters experience in an imaginary fashion, as the holistic projection of the unconscious notion.

However the concept of number then becomes quickly reduced with respect to interpretation in a real rational fashion (as applying to all actual numbers).

From the opposite perspective, in the movement from the concept of number to the number "2", initially a holistic unconscious aspect is involved, where one intuits the potential notion of number as being embodied in the number "2" (in imaginary fashion). However, this again becomes quickly reduced to the actual interpretation of "2" (in a real conscious manner).

Thus we have now identified in experience, four poles with respect to the number "2", whereby it is identified in both a real (analytic) and imaginary (holistic) fashion with respect to both perceptual and conceptual directions. And in experience, one (consciously) posits perceptions by (unconsciously) negating concepts; equally one posits concepts by negating perceptions

So, we have identified in a qualitative holistic manner, two real poles and two imaginary poles that are positive and negative with respect to each other.

Put another way, we have identified "2" as a holon, which dynamically is identified with all four poles of the complex plane (that in geometrical terms divides the unit circle into four equal quadrants).

In conventional mathematical terms, these four directions (i.e. dimensions) are simply reduced in a linear (1-dimensional) fashion. In other words "2" is treated as an independent existing number in an absolute conscious manner.

Now mathematicians, if pressed might concede that we cannot have knowledge of number (in the absence of the number concept). However they are unlikely to be interested in such "philosophical" digressions and will just carry on treating number perceptions in a reduced independent fashion. And as for any appreciation that number may entail inescapable holistic considerations (of an unconscious nature), this is unlikely to ever receive consideration!

We have now identified four fundamental polarities (internal/external and whole/part) that fundamentally underlie all phenomenal experience (including as I have illustrated, mathematical).

Furthermore, we have shown that these four poles interact in a dynamic manner, that is perfectly represented through the holistic mathematical interpretation of the four roots of 1.

So for every phenomenon, we have now literally defined a new 4-dimensional holistic framework based on twin real (analytic) complementary polarities, that are positive and negative with respect to each other and likewise twin imaginary (holistic) complementary polarities, that are likewise positive and negative with respect to each other.

These equally apply to all physical and all psychological relationships (which are themselves now seen as complementary).

This scientific understanding (of a qualitative kind), which is very much representative in my framework of development with Band 3 (Level 2), has far reaching consequences.

Properly understood therefore, we cannot, in any context, physical or psychological, explain the dynamic relationship of wholes and parts in the absence of holistic appreciation (of a direct unconscious nature). However, indirectly, this holistic aspect can be incorporated in an imaginary rational manner.

So from this new perspective, all reality is now scientifically understood in a qualitative complex rational manner (comprising real and imaginary aspects).

So just as the imaginary notion has now a fully accepted quantitative interpretation in Conventional Mathematics, equally the imaginary notion has an - as yet unrecognised - qualitative interpretation that is inseparable from the proper dynamic appreciation of all phenomenal relationships.

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